Unbiased Estimates for Products of Moments and Cumulants for Finite Populations

Let FN be the distribution function of a finite real population of size N. Let Fn be the empirical distribution function of a sample of size n drawn from the population without replacement. Let TFN be any product of the moments or cumulants of FN, let TFn denote the sample version, and let Tn,NFN denote the expected value of TFn with respect to FN. We prove the following remarkable inversion principle that the expected value of TN,nFn is equal to TFN. We also obtain an explicit expression for Tn,NFN for all TFN of orders up to six.